32 Chapter 1. Sets, Real Numbers and Inequalities
(a) A−B= A0∩B
(b) (A∪B)∩C=A∪(B∩C) (c) (A0∪B0)∩B=B−A
A statement above is true means that it is true for all possible choices of A, B, C and U. To show that the statement is false, it is enough to give a counterexample. To show that it is true, you can draw a Venn diagram to convince yourself; but to be more rigorous, you should use formal mathematical logic.
1.2. Real Numbers 33
InR, we have thealgebraic operations+,×(and−,÷also) as well asbinary relations<,≤, >,≥. Numbers greater than (respectively smaller than) 0 are calledpositive(respectivelynegative).
Real Number Line Real numbers can be represented by points on a line, called thereal number line.
>
|
−1
|
0
|
1
|
2 Figure 1.4
Notation The following nine types of subsets ofRare calledintervals:
[a,b] = {x∈R:a≤ x≤b} (1.2.1)
(a,b) = {x∈R:a< x<b} (1.2.2)
[a,b) = {x∈R:a≤ x<b} (1.2.3)
(a,b] = {x∈R:a< x≤b} (1.2.4)
[a,∞) = {x∈R:a≤ x} (1.2.5)
(a,∞) = {x∈R:a< x} (1.2.6)
(−∞,b] = {x∈R:x≤b} (1.2.7)
(−∞,b) = {x∈R:x<b} (1.2.8)
(−∞,∞) = R (1.2.9)
whereaandbare real numbers witha<band∞and−∞(read “infinity” and “minus infinity”) are just symbols but not real numbers.
FAQ What are the meaning of∞and−∞?
Answer Intuitively, you may imagine that there is a point, denoted by∞, very far away on the right (and−∞
on the left). So (a,∞) is the set whose elements are the points betweenaand∞, that is, real numbers greater
thana.
Remark The notation (a,b), wherea<b, has two different meanings. It denotes an ordered pair as well as an interval. To avoid ambiguity, some authors use ]a,b[ to denote the open interval{x ∈R :a < x < b}. In this course, we will not use this notation. Readers can determine the meaning from the context.
Terminology
• Intervals in the form (a,b), [a,b], (a,b] and [a,b) are called bounded intervalsand those in the form (−∞,b), (−∞,b], (a,∞), [a,∞) and (−∞,∞) are calledunbounded intervals.
• Intervals in the form (a,b), (−∞,b), (a,∞) and (−∞,∞) are called open intervals. For each of such intervals, the endpoint(s), if there is any, does not belong to the interval.
• Intervals in the form [a,b], (−∞,b], [a,∞) and (−∞,∞) are calledclosed intervals. For each of such intervals, the endpoint(s), if there is any, belongs to the interval.
• Intervals in the form [a,b] are calledclosed and bounded intervals.
34 Chapter 1. Sets, Real Numbers and Inequalities
• A set{a}with exactly one element ofRis called adegenerated interval(its length is 0).
• Some authors also include∅as an interval (called theempty interval).
In this course, an interval means a nonempty, non-degenerated interval, that is, an infinite subset ofRthat can be written in the form (1.2.1), (1.2.2), (1.2.3), (1.2.4), (1.2.5), (1.2.6), (1.2.7), (1.2.8) or (1.2.9).
Example For each of the following pairs of intervalsAandB, (1) A=[1,5] andB=(3,10]
(2) A=[−2,3] andB=(7,11]
(3) A=[−7,−2) andB=[−2,∞)
• determine whether it is (i) an open interval, (ii) a closed interval , (iii) a bounded interval;
• findA∩Band determine whether it is an interval.
• findA∪Band determine whether it is an interval.
Solution
(1) BothAandBare not open intervals.
Ais a closed interval butBis not a closed interval.
BothAandBare bounded intervals.
A∩B=(3,5]; it is an interval.
A∪B=[1,10]; it is an interval.
(2) BothAandBare not open intervals.
Ais a closed interval butBis not a closed interval.
BothAandBare bounded intervals.
A∩B=∅; it is not an interval.
A∪B=[−2,3]∪(7,11]; it is not an interval.
(3) BothAandBare not open intervals.
Bis a closed interval butAis not a closed interval.
Ais a bounded interval butBis not a bounded interval.
A∩B=∅; it is not an interval.
A∪B=[−7,∞); it is an interval.
1.2.2 Radicals Definition
(1) Letaandbbe real numbers and letqbe a positive integer. Ifaq =b, we say thatais aqth rootofb.
Example
(a) −2 is the cube root of−8.
(b) 3 and−3 are the square roots of 9.
1.2. Real Numbers 35
Note
(a) Ifqis odd, then every real number has a uniqueqth root.
(b) Ifqis even, then
(i) every positive real number has twoqth roots;
(ii) negative real numbers do not haveqth root;
(iii) theqth root of 0 is 0.
(2) Letbbe a real number and letq be a positive integer. The principal qth rootofb, denoted by √q b, is defined as follows:
(a) ifqis odd,√q
bis the uniqueqth root ofb;
(b) ifqis even, (i) √q
bis the positiveqth root ofbifb>0;
(ii) √q
bis undefinedbifb<0;
(iii) √q
bis 0 ifb=0.
Whenq=2, √2
xis simply written as √ x.
FAQ Can we write√
4=±2 ? Answer According to the definition,√
4 is the principle square root of 4, which is the positive real number whose square is 4. That is,√
4=2.
FAQ In solvingx2=4, we getx=±2. Is this different from the above question?
Answer To find√
4 is different from solvingx2=4.
(a) √
4 is a uniquely defined real number.
(b) To solvex2 = 4 is to find real numbers whose square is 4. There are two such numbers, namely 2 and−2.
Don’t mix up the two questions.
Example (a) √4
81=3 (b) √3
−8=−2 (c) √
25= √2 25=5 (d) √
0= √2 0=0 (e) √6
−3 is undefined.
Terminology The symbol √q
bis called aradical(qis called theindexandbtheradicand).
FAQ Is√
a2=aalways true?
Answer It is true if (and only if)a≥0. Ifa<0, we have √
a2=−a.
36 Chapter 1. Sets, Real Numbers and Inequalities
(3) Letbbe a positive real number. Letpandqbe integers whereq>0. We define bpq = √q
bp, which is the same as√q
bp . Example 823 = √3
82 = √3 64=4 RemarkEquivalently, we have 823 =√3
82
=22 =4.
FAQ Are the rules for exponents on page 1 valid ifmandnare rational numbers?
Answer The rules remain valid for rational exponents, provided that the base is positive (this is required in the definition ofbqp). For example, we havebsbt =bs+t, whereb>0 ands,t∈Q.
Proof Writes= m
n andt= p
q wherem,n,p,qare integers withq,n>0. Note that s= mq
nq, t= np
nq and s+t= mq+np
nq . By definition (equivalent form), we have
bs=nq√ bmq
and bt =nq√ bnp
. Denoteα= nq√
b. Then we have
bs·bt = αmq·αnp
= αmq+np
= αnq(s+t)
= αnqs+t
= bs+t
FAQ Can we definebraising to an irrational power? For example, can we define 2π? How?
Answer This is deep question. The idea will be discussed Chapter 8.
Exercise 1.2
1. Find the following sets.
(a) {x∈R:x2 =2}
(b) {x∈R:x≥0 andx2=2}
(c) {x∈Q:x2 =2}
2. LetA=[1,5],B=[3,9),C={1,5}andD=[5,∞). Find
(a) A∩B (b) A∪B
(c) A−C (d) B∩C
(e) C−B (f) B−C
(g) B−(B−C) (h) A∪D
(i) C∩D